Power series are like building blocks for functions, letting us represent complex mathematical relationships as infinite sums. They're super useful for approximating functions and solving tricky equations. Understanding power series is key to grasping advanced calculus concepts.
In this section, we'll dive into the nuts and bolts of power series. We'll learn how to define them, identify their parts, and perform basic operations. Plus, we'll explore how to determine when and where these series converge.
Power Series Structure
Definition and Representation
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Power series are infinite series of the form ∑n=0∞an(x−c)n, where:
an are the coefficients that determine the magnitude of each term
c is the center, which represents the point around which the series is expanded
x is the variable
Power series represent functions as an infinite sum of terms involving powers of a variable
Examples of functions that can be represented by power series include polynomials (x2+2x+1), exponential functions (ex), and trigonometric functions (sin(x), cos(x))
General Term and Domain of Convergence
The general term of a power series is an(x−c)n, where n is a non-negative integer
This term determines the structure of each individual component in the infinite series
The domain of convergence of a power series is the set of all values of x for which the series converges
Outside the domain of convergence, the series may diverge or have undefined behavior
The domain of convergence can be determined using various convergence tests
Center and Coefficients of Power Series
Identifying the Center
The center of a power series is the value c in the general term an(x−c)n
It represents the point around which the series is expanded
To identify the center, look for the value of x that is being subtracted from the variable in each term
For example, in the series ∑n=0∞(x−3)n, the center is c=3
Identifying the Coefficients
The coefficients of a power series are the values an in the general term an(x−c)n
They determine the magnitude of each term in the series
To identify the coefficients, look for the constant values multiplied by each power of (x−c)
For example, in the series ∑n=0∞2n(x−1)n, the coefficients are an=2n
The center and coefficients can be used to determine properties such as the radius of convergence and behavior near the center
Arithmetic Operations with Power Series
Addition of Power Series
Power series with the same center can be added term by term, provided that the resulting series converges
To add two power series, add the coefficients of like powers of (x−c)
For example, ∑n=0∞an(x−c)n+∑n=0∞bn(x−c)n=∑n=0∞(an+bn)(x−c)n
The resulting power series will have the same center as the original series
Multiplication of Power Series
Power series can be multiplied using the Cauchy product
Multiply each term of one series by each term of the other and collect like powers of (x−c)
The Cauchy product of two power series ∑n=0∞an(x−c)n and ∑n=0∞bn(x−c)n is given by ∑n=0∞cn(x−c)n, where cn=∑k=0nakbn−k
The resulting power series will have the same center as the original series
The radius of convergence of the product will be at least the minimum of the radii of convergence of the original series
Convergence of Power Series
Convergence at a Specific Point
A power series converges at a specific point x=x0 if the sequence of partial sums Sn(x0)=∑k=0nak(x0−c)k approaches a finite limit as n approaches infinity
Convergence at a specific point can be determined using various tests:
Ratio test: If limn→∞∣an(x0−c)nan+1(x0−c)n+1∣<1, the series converges at x=x0. If the limit is greater than 1, the series diverges at x=x0
Root test: If limn→∞n∣an(x0−c)n∣<1, the series converges at x=x0. If the limit is greater than 1, the series diverges at x=x0
Comparison test: Compare the terms of a power series with the terms of a known convergent or divergent series to determine convergence
Convergence Behavior
If a power series converges at a specific point, it will converge for all points closer to the center
If a power series diverges at a specific point, it will diverge for all points farther from the center
The set of all points where a power series converges is called the interval of convergence
The interval of convergence is always centered around the center of the power series
The radius of convergence is the distance from the center to the endpoints of the interval of convergence